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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_4.html ./knowl/eq3_5.html ./knowl/eq3_5.html">
\begin{equation}
y^{\prime \prime}+b y^{\prime}+c y=0.\tag{3.1.6}
\end{equation}
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<p class="continuation">Seek a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_4.html ./knowl/eq3_5.html ./knowl/eq3_5.html">
\begin{equation}
y=e^{rx},\tag{3.1.7}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\(r\)</span> needs to be determined. Substituting (<a href="" class="xref" data-knowl="./knowl/eq3_4.html" title="Equation 3.1.7">(3.1.7)</a>) into (<a href="" class="xref" data-knowl="./knowl/eq3_5.html" title="Equation 3.1.6">(3.1.6)</a>):</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_4.html ./knowl/eq3_5.html ./knowl/eq3_5.html">
\begin{equation*}
r^2 e^{rx}+bre^{rx}+ce^{rx}=0~\rightarrow~r^2+br+c=0.
\end{equation*}
</div>
<p class="continuation">This algebraic equation for <span class="process-math">\(r\)</span> is called the <dfn class="terminology">characteristic equation</dfn> of (<a href="" class="xref" data-knowl="./knowl/eq3_5.html" title="Equation 3.1.6">(3.1.6)</a>).</p>
<span class="incontext"><a href="sec3_1.html#p-63" class="internal">in-context</a></span>
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